A Chaos theory

Chaos could appear in the system if there is some nonlinearity. My question is how to explain that there is no chaos in constant acceletating system
[tex]s(t)=v_0t+\frac{at^2}{2}[/tex]
when equation is nonlinear? Why is important only that difference and differential equation be nonlinear. It confusing me.
 

fresh_42

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Chaos could appear in the system if there is some nonlinearity. My question is how to explain that there is no chaos in constant acceletating system
[tex]s(t)=v_0t+\frac{at^2}{2}[/tex]
when equation is nonlinear? Why is important only that difference and differential equation be nonlinear. It confusing me.
The crucial point is probably, that ##s(t)## isn't sensitive on the initial conditions. To answer why, I'll have to ask you which definition of sensitivity do you want to use (I've found three on Wikipedia). I assume the entire trajectory will be the underlying topological space as well as its dense ##s-##invariant subset in order to clear the topological assumptions.
 
Exponential sensitivity. Calculation of largest Lyapunov exponent.
 

fresh_42

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A back of the envelope calculation with the first differential as quotient of rates, the formula ##\log (1+t) = - \log (1 - (\frac{t}{1+t}))## and the power series of ##\log## gave me a Lyapunov exponent zero.
 

hilbert2

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An anharmonic 1D oscillator with Lagrangian ##\frac{p^2}{2m}-kx^4## isn't chaotic either, even though the equation of motion is nonlinear in ##x(t)##. On the other hand, you can make a harmonic oscillator chaotically sensitive to the initial condition by choosing a negative "spring constant" (a slightest deviation from the equilibrium position will start growing exponentially) . Note that in the constant accelerating system the force is an s-independent constant, which means that it's actually a mechanical system with a linear equation of motion even though the trajectory is a nonlinear function of ##t##.
 
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